dot-lagrangian_project_woodbury: Woodbury-Corrected Lagrangian Projection

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Woodbury-Corrected Lagrangian Projection

Description

Applies the constrained projection for the structured-correlation Woodbury path. In supplement notation, \mathbf{G}^{1/2} = \mathbf{G}_{\mathrm{on}}^{1/2}\mathbf{F}^{1/2}, where \mathbf{G}_{\mathrm{off}}^{-1} = \mathbf{E}\mathbf{J}\mathbf{E}^{\top} and \mathbf{F}^{1/2} carries the low-rank correction.

Usage

.lagrangian_project_woodbury(
  GhalfXy_V,
  Ghalf_corrected,
  A,
  K,
  p_expansions,
  R_constraints,
  constraint_value_vectors,
  family,
  wb_sqrt,
  parallel_aga,
  parallel_matmult,
  cl,
  chunk_size,
  num_chunks,
  rem_chunks,
  parallel_qr = FALSE
)

Arguments

GhalfXy_V	Numeric column vector of length P; the fully transformed right-hand side \mathbf{G}^{1/2}\mathbf{X}^{\top}\mathbf{V}^{-1}\mathbf{y} assembled through the Woodbury-corrected operator.
Ghalf_corrected	List of K+1 corrected \mathbf{G}_{\mathrm{on},k}^{1/2} matrices.
A	Constraint matrix (P \times R).
K, p_expansions, R_constraints	Integer dimensions.
constraint_value_vectors	Constraint RHS list encoding \mathbf{A}^{\top}\boldsymbol{\beta} = \mathbf{c}.
family	GLM family object.
wb_sqrt	Output of .woodbury_halfsqrt_components, containing the internal rank-r representation of \mathbf{F}^{1/2}: Q (basis matching supplement \mathbf{Q}), C and C_inv (diagonal correction coefficients), and GhalfQ (partition-wise copies of \mathbf{G}_{\mathrm{on},k}^{1/2}\mathbf{Q}[rows_k,]).
parallel_aga, parallel_matmult	Logical flags.
cl, chunk_size, num_chunks, rem_chunks	Parallel parameters.

Details

The routine follows the same four transformed-OLS steps as .lagrangian_project: form \mathbf{y}^{*}, form \mathbf{X}^{*}, solve the OLS projection, then back-transform. Only steps involving \mathbf{G}^{1/2} differ, and those are handled as a block-diagonal multiplication through \mathbf{G}_{\mathrm{on}}^{1/2} plus a rank-r update.

Internally, \mathbf{F}^{1/2} is stored as \mathbf{I} - \mathbf{Q}\mathbf{C}\mathbf{Q}^{\top}, where Q is the basis corresponding to the supplement matrix \mathbf{Q} and C is diagonal.

Value

A list of K+1 coefficient column vectors.

lgspline documentation built on Aug. 5, 2026, 1:10 a.m.